Interval Estimation for Quantile on Two Parameters Exponential Distribution Under Multiple Type-II Censoring on Complex Case

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Jurnal ILMU DASAR, Vol. 10 No. 1, Januari 2009 : 13 - 17 13

Interval Estimation for Quantile on Two Parameters Exponential Distribution Under Multiple Type-II Censoring on Complex Case

with Bootstrap Percentile

Akhmad Fauzy

Department of Statistics, Universitas Islam Indonesia, Yogyakarta ABSTRACT

In this article, two methods are proposed to give the interval estimation for quantile on two parameters exponential distribution under multiply type II censoring. The interval estimation for quantile can be constructed the estimated parameters. Those researchers have use approximate maximum likelihood estimator to construct interval estimation for two parameters exponential distribution under multiply type II censoring. All of these method need an assumption that sample is exponentially distributed. We will use another method, known as the bootstrap percentile. This method gives shorter interval than the traditional method and this method does not need an assumption that the sample is distributed exponentially.

Keywords: bootstrap percentile, multiply type II censoring, quantile

INTRODUCTION

The exponential distribution has immensely contributed in the analysis of lifetime.

Historically the exponential distribution was the first lifetime model in which statistical methods in survival analysis were extensively developed. What distinguishes survival analysis from other fields of statistics is censoring. Vaguely speaking, a censored observation contains only partial information about the random variable of interest. The three types of censoring are the type-I censoring, the type-II censoring and random censoring (Kleinbaum & Klein 2005).

Balakrishnan, Kundu & Kannan (2007), Kang (2003), Balasubramanian & Balakrishnan (1992) and Balakrishnan (1990) use approximate maximum likelihood estimator to construct point estimation for one parameter exponential distribution under multiple type-II censoring on complex case. Fei & Kong (1994) use traditional method to construct interval estimation for one parameter exponential distribution under multiple type-II censoring on complex case. This interval needs an assumption that sample is exponentially distributed. Bootstrap method is a computer- based method for assigning measures of accuracy to statistical estimates, especially to calculate the confidence interval. The aim of using bootstrap method is to gain the best estimation from minimal data (Efron &

Thibshirani 1993).

Fauzy et al. (2002a, 2002b, 2003a, 2003b, and 2005) used bootstrap method to construct interval estimation for quantile on one and two parameters exponential distribution under single type II censoring, double type II censoring and multiple type II censoring on simple case. In this paper comparison study will be made on the interval estimation for quantile on two parameters exponential distribution under multiply type II censoring on complex case using both the conventional method and bootstrap percentile method.

METHODS

The data used for illustration is taken from Balasubramanian & Balakrishnan (1992), Fei &

Kong (1994) and Kang (2003). We begin with the interval estimation for quantile on two parameters exponential distribution under multiple type-II censoring on complex case using the traditional method. This will be followed with the searching of convergence condition from bootstrap’s repeated samples. Next, after knowing the convergence condition, confidence interval for quantile on two parameters exponential distribution under multiple type-II censoring on complex case with bootstrap percentile method will be computed.

Two parameter exponential distribution

The two parameters exponential distribution has probability density function (Lawless 2003):

; η, θ exp ; η, θ 0 (1)

Here η is the warranty time and θ is the residual mean life. Suppose n items are placed on a life-

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14 Interval Estimation ... (Akhmad Fauzy)

testing experiment and that the th, th, , th failure times ate only made available. That is,

: , _: , , _: (2)

is the multiply type-II censored sample available from (1).

The likelihood function based on the multiply type- II censoring on complex case in (2) is given by:

const. 1 exp exp

1 exp ∑ exp

3 From (3) the likelihood equation for θ is obtained as:

ln θ

1

θ θ 1

1 exp θ exp θ

exp θ exp θ

(4) The estimation for θ is (Balasubramanian and Balakrishnan, 1992):

( )( ( ) ) ( ) ( )

( )

∑ − − γ

−

∑ +∑ − − δ + −δ + − − − +

= θ

= +

1 -

1 1

1 1 1 1

1 -

1 1

1

1 1

ˆ 1

k

i i i i

k

i ii i i k k

k

i i i

i

r r k

t r n t r n t t r r t

(5) The (1- α) lower confidence bound for θ is (Fei and Kong, 1994):

( )

⎟⎟

⎠

⎞

⎜⎜

⎝ χ ⎛ μ

⎥⎦⎤

⎢⎣⎡∑ − μ

=

⎟⎟

⎠

⎞

⎜⎜

⎝ χ ⎛ μ

⎥⎦⎤

⎢⎣⎡∑ − − μ

= θ

α

−

= α

−

=

1 : 2 1 : 12 1 :

1: 1 : 1 :

1 : 2 1 : 12 1 :

1: 2 :

1 :

min 2

4 2

1 ˆ 4

k k k

n r k i rin k

k k k

n r k

i rin k

V V

t k t

V V

t k t

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where _: 2 ∑ ∑ and _:

4 ∑ ∑

For the case when r1 = 1, we simply observe from (3) that L is a monotonic increasing function in η and hence the maximum likelihood estimator for η is

simply ̂ : : ..

For the case when r1 > 1 the likelihood equation for η:

( ) ( )

0

eksp 1

eksp 1 1 1

ln

1 1 1

1 =

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛ θ

η

− −

−

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ ⎟

⎠

⎜ ⎞

⎝

⎛ θ

η

− −

−

− + θ − η =

∂

t t r

r L n

(7) yields the maximum likelihood estimator for η to be:

⎟⎠

⎜ ⎞

⎝

⎛ − +

θ +

=

η n

r

t n 1

ˆln

ˆ ₁ ¹

(8) The (1- α) lower confidence bound for η is (Fei and Kong, 1994):

( )

_⎢⎣⎡∑ − _⎥⎦^⎤

μ

− μ

=

η −α =

k

i irn r n

k n

r F f f t kt

t 1 1 2 1 : 1:

1 : : 1

min 1 ,

ˆ

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where and ^:

The pth quantile on one parameter exponential :

distribution, given by tp, and so:

(

−

)

θ

− μ

= ˆ ln1 p ˆ

t_p (10) The (1-α) confidence for survivor function is:

( )

_min _max

( )

_max

min log1 ˆ ˆ log1 ˆ

ˆ − − θ 〈 〈 μ − − θ

μ p t_p p

(11) Bootstrap percentile method

In setting up of the bootstrap method to find the confidence intervals and estimating significance levels, the method consists of approximating the distribution of a function of the observations and the underlying distribution, such as the pivot, denoted by Efron as the bootstrap distribution of this quantity. This distribution is obtained by replacing the unknown distribution by the empirical distribution of the data in the definition of the statistical function, and then resampling the data to obtain a Monte Carlo distribution for the resulting random variable (Chernick 1999).

Bootstrap method is a computer-based method for assigning measures of accuracy to statistical estimates, especially to calculate the confidence interval. Bootstrap itself comes from the phrase “pull oneself up by one’s Bootstrap” which means to stand up by one’s own feet and do with minimal resources.

The minimal resource is a minimum data, data which are free from certain assumption or data with no assumption at all about the population distribution.

The aim of using bootstrap method is to gain the best estimation from minimal observation (Chernick 2007).

The Bootstrap’s percentile procedures for the interval estimation for quantile on two parameters exponential distribution under multiple type-II censoring on complex case are as follows:

1. give an equal opportunity to every observation, 2. take n sample with replication,

3. do step 2 until B times in order to get an

“independent bootstrap replications”, , , … , and search for convergence condition. Calculate:

( ) ^b

b b

p p

t^∗ = μˆ^∗ −ln1− θˆ^∗ with (12)

⎟⎠

⎜ ⎞

⎝

⎛ − + θ

+

=

η^∗ ^∗ ^∗

n r ln n t ^b_n ^b

b ˆ 1

ˆ ₁_: ¹

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b n b

r n b

b t^∗ x^∗ x^∗

∗ = = =

η ₁_:

: 1 1

ˆ ^for^r⁼¹

( ) ( ( ) ) ^{( ) (} ⁾

( )

∑ − − γ

−

∑ +∑ − − +δ + −δ + − − − +

= θ

= +

=

∗

∗ +

∗

= +

∗

1 -

1 1

1 1 1 1

1 -

1 1

1

1 1

ˆ 1

k

i i i i

k i

b b

k k b i i b ii k

i i i

b b i

r r k

t r n t r n t t r r t

4. define the confidence interval at the level (1-α) of the bootstrap percentile for quantile on two parameters exponential distribution under multiple type-II censoring on complex case as:

( ) ( )

[

B /² ^, p^B ^/²

]

p t

t^* ^α ^* ¹⁻^α

(13) Expressions (13) refer to the ideal bootstrap

situation where the bootstrap replications are infinite. So if B = 2000 and α = 0.05, is the 50th and is the 1950th ordered value of the replications.

RESULTS AND DISCUSSION

We consider the data presented in Balasubramanian and Balakrishnan (1992), Fei and Kong (1994) and Kang (2003). It is a simulated lifetime data set (in hours) from two parameters exponential distribution with θ = 20.

The data is given as follow:

0.961 0.990 1.565 2.031 2.204 2.340 3.642 6.008 6.538 7.145

- - - 11.937 15.433

18.234 18.307 22.096 - - - 28.799 30.692 30.737 33.702

34.245 - - - -

We will construct interval estimation for 0.30 and 0.70 quantile (t0,30 and t0,70). From equations (10) for traditional method and (12) for bootstrap methods, the point estimates of:

t0,30= 7.680639 and , = 8.678485 t0,70 = 23.643450 and , = 26.222990 Based on the above under multiple type-II censoring on complex case, the intervals estimation for 0.30 quantile in Table 1 and for 0.70 quantile in Table 2. Bootstrap’s repeated result gives a convergence condition that begins at B = 4000 for 0.30 quantile and B = 4000 for 0.70 quantile. The plot between bias and replication are shown in Figure 1 for 0.30 quantile and Figure 2 for 0.70 quantile.

Table 1. The floor (F), ceiling (C) and interval widths (IW) for 0.30 quantile at the level of (L) 99 % and 95 %.

L Traditional method Bootstrap method

F C IW F C IW

99 % 1.28015 13.01153 11.73138 5.37775 12.39715 7.01940 95 % 3.04691 11.37413 8.32722 6.11508 11.41834 5.30326

Table 2. The floor (F), ceiling (C) and interval widths (IW) for 0.70 quantile at the level of (L) 99 % and 95 %.

L Traditional method Bootstrap method

F C IW F C IW

99 % 11.37994 41.64601 30.26607 15.86223 36.97318 21.11095 95 % 14.28626 36.15094 21.86468 18.30130 34.08008 15.77878

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9000 8000 7000 6000 5000 4000 3000 2000 1000 0 1.1

1.0

0.9

0.8

0.7

replication

bias

9000 8000 7000 6000 5000 4000 3000 2000 1000 0 2.8

2.3

1.8

replication

bias

Figure 1. Plot between replication and bias 0.30 quantile.

Figure 2. Plot between replication and bias 0.70 quantile.

Table 3. Comparison interval widths for 0.10 and 0.20 quantile at level of 99 % and 95 %.

Method

30 .

t

0

t

₀_.₇₀

99 % 95 % 99 % 95 %

Traditional 11.73138 8.32722 30.26607 21.86468

Bootstrap percentile 7.01940 5.30326 21.11095 15.77878

Difference interval 4.71198 3.02396 9.15512 6.08590

Comparison of interval widths

Table 3 gives the interval widths of the 0.30 and 0.70 quantile on two parameters exponential distribution under multiple type-II censoring on complex case using the traditional method and the bootstrap percentile method

CONCLUSION

Bootstrap percentile method is a more potential method in constructing interval estimation for quantile on two parameters exponential

distribution under multiple type-II censoring on complex case than the traditional method whereby it gives a shorter interval. And this method does not need an assumption that the sample has to have an exponential distribution.

REFERENCES

Balakrishnan. 1990. On the Maximum Likelihood Estimation of the Location and Scale Parameters of the Exponential Distribution Based on Multiply Type-II Censored Samples. Journal of

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18 Interval Estimation ... (Akhmad Fauzy)

Balakrishnan, Kundu HKT, Kannan N. 2007. Point and Interval Estimation for a Simple Step-Stress Model with Type-II Censoring. Journal of Quality Technology 3:935-47

Balasubramaniam K & Balakrishnan N. 1992.

Estimation for One and Two Parameter Exponential Distributions under Multiple Type- II Censoring. Statistical Papers 33:203-216.

Chernick MR. 1999. Bootstrap Methods: A Practitioner's Guide. New York: John Wiley &

Sons.

Chernick MR. 2007. Bootstrap Methods: A Practitioners and Researchers. New York:

Wiley Interscience.

Efron B & Tibshirani R. 1993. An Introduction to the Bootstrap. New York: Chapman & Hall.

Fauzy A, Ibrahim NA, Daud I, Abu Bakar MR.

2002a. Interval Estimation for Quantile on One Parameter Exponential Distribution under Type II Censoring with Bootstrap Percentile.

Proceedings National Conference on Mathematics, Department of Mathematics, Institute of Technology Surabaya, 2 November 2002.

2002b. Interval Estimation for Quantile on Two Parameters Exponential Distribution under Type II Censoring with Bootstrap Percentile.

Proceedings National Conference on Mathematics, Department of Mathematics, Institute of Technology Surabaya, 2 November 2002

2003a. Interval Estimation for Quantile on Exponential Distribution under Double Type II Censoring with Bootstrap Percentile. Poster of the International Conference on Reliability and Survival Analysis 2003. University of South Carolina US, 21-24 May 2003.

Fauzy A, Ibrahim NA., Daud I, Abu Bakar MR.

2003b. Interval Estimation for One and Two Parameters Exponential Distribution under Multiple Type-II Censoring with Bootstrap Percentile. Proceedings of the Regional Conference on Integrating Technology in the Mathematical Sciences 2003. April 14-15 2003.

Universiti Sains Malaysia.

2005. Interval Estimation for Parameters Exponential Distribution under Double type-II censoring. IRSIAM Journal on Applied Mathematics (IJAM) 1:81-95.

Fei H & Kong F. 1994. Interval Estimations for One and Two Parameters Exponential Distributions under Multiple Type II Censoring. Commun.

Statist.-Theory Meth. 23(6):1717-1733.

Kang SB. 2003. Approximate MLE for Exponential Distribution under Multiple Type-II Censoring.

Journal of Korean Data & Information Science Society 14:983-988.

Kleinbaum DG & Klein. 2005. Survival Analysis.

2^nd ed. New York: Springer.

Lawless. 2003. Statistical Models and Methods for Lifetime Data. 2^nd ed. New York: John Wiley &

Sons.

Interval Estimation for Quantile on Two Parameters Exponential Distribution Under Multiple Type-II Censoring on Complex Case